Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Charles E Chidume,Abubakar Adamu,Lois C Okereke

doi:10.1186/s13663-020-0670-7

Charles E Chidume, Abubakar Adamu + Show 1 more

Open Access

https://doi.org/10.1186/s13663-020-0670-7

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Abstract

Let B be a uniformly convex and uniformly smooth real Banach space with dual space B^{*}. Let F:Bto B^{*}, K:B^{*} to B be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u+KFu=0. This theorem is a significant improvement of some important recent results which were proved in real Hilbert spaces under the assumption that F and K are maximal monotone continuous and bounded. The continuity and boundedness restrictions on K and F have been dispensed with, using our new method, even in the more general setting considered in our theorems. Finally, numerical experiments are presented to illustrate the convergence of the sequence of our algorithm.

Highlights

Let B be a real Banach space with a strictly convex dual space B∗
The first iterative methods for approximating solutions of Hammerstein equations, in real Banach spaces more general than Hilbert spaces, as far as we know, were obtained by Chidume and Zegeye [25]
Chidume and Zegeye [25] were able to prove strong convergence of an iterative algorithm defined in the Cartesian product space E to a solution of the Hammerstein equation (1.1)

Summary

Introduction

Let B be a real Banach space with a strictly convex dual space B∗. Consider the Hammerstein equation (I + KF)u = 0, (1.1)where F : B → B∗ is a nonlinear mapping and K : B∗ → B is a linear map such that R(F) ⊂ D(K). The first iterative methods for approximating solutions of Hammerstein equations, in real Banach spaces more general than Hilbert spaces, as far as we know, were obtained by Chidume and Zegeye [25] We note that T[u, v] = 0 ⇐⇒ u solves (1.1) and v = Fu. With this, Chidume and Zegeye [25] were able to prove strong convergence of an iterative algorithm defined in the Cartesian product space E to a solution of the Hammerstein equation (1.1).

Results

Conclusion